Symbiotic comma: Difference between revisions

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'''19712/19683''', the '''symbiotic comma''', also known as the '''symbiosis''' or '''symbiosma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup]]) [[comma]] measuring about 2.55 [[cent]]s. It is the difference between an [[2187/2048|apotome (2187/2048)]] and an [[77/72|undecimal secor (77/72)]] – the latter of which can be broken down into a quartertone of [[33/32]] stacked on top of a third-tone of [[28/27~~]]. It is also the difference between a [[septischisma]] and a [[olympia~~]].

'''19712/19683''', the '''symbiotic comma''', also known as the '''symbiosis''' or '''symbiosma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup]]) [[comma]] measuring about 2.55 [[cent]]s. It is the difference between a [[2187/2048|Pythagorean apotome (2187/2048)]] and an [[77/72|undecimal secor (77/72)]] – the latter of which can be broken down into a quartertone of [[33/32]] stacked on top of a third-tone of [[28/27]].

~~It factors into~~ the ~~following commas:~~

In terms of other commas, it is the difference between a [[septischisma]] and an [[olympia]]. It is also the sum of an [[argyria]] and an olympia, as well as the sum of a [[quartisma]] and a [[nexus comma]]. In the full 11-limit, it factors into simpler commas as ([[4375/4374]])⋅([[5632/5625]]), and in the 19-limit, ([[1216/1215]])⋅([[1540/1539]]).

*[[~~2.3.7.11 subgroup~~]]:

~~:* (~~[[olympia]]~~)⋅(~~[[argyria]])

~~:* (~~[[quartisma]]~~)⋅(~~[[nexus comma]])

* [[11-limit~~]]:~~

~~:* ([[3025/3024]])⋅([[reef comma]])~~

:*([[4375/4374]])⋅([[5632/5625]])

* [[13-limit]]

~~:* ([[676/675]])⋅([[chalmersia]])~~

~~:* ([[1001/1000]])⋅([[phaotic comma]])~~

~~:* ([[4096/4095]])⋅([[35035/34992]])~~

* [[2.3.5.7.11.13.19 subgroup]]

:* ([[1216/1215]])⋅([[1540/1539]])

== Temperaments ==

[[Tempering out]] this comma in the 11-limit results in the rank-4 '''symbiotic''' temperament (→ [[Rank-4 temperament #Symbiotic (19712/19683)]]), or in the 2.3.7.11 subgroup, the '''symbiotian''' temperament. In either case, ~~it identifies~~ [[11/8]] by an apotome over [[9/7]], [[11/6]] by an apotome over [[12/7]], and [[11/9]] by an apotome over [[8/7]]. Finally, ~~it identifies~~ 11/9 by a [[256/243|limma]] below 9/7.

[[Tempering out]] this comma in the 11-limit results in the rank-4 '''symbiotic''' temperament (→ [[Rank-4 temperament #Symbiotic (19712/19683)]]), or in the 2.3.7.11 subgroup, the '''symbiotian''' temperament. In either case, 28/27 and 33/32 become apotome complements, and [[11/8]] is identified with an apotome over [[9/7]], [[11/6]] with an apotome over [[12/7]], and [[11/9]] with an apotome over [[8/7]]. Finally, 11/9 is identified with a [[256/243|limma]] below 9/7.

=== Symbiotian ===

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[[Comma list]]: 19712/19683

: mapping generators: ~2, ~3, ~7

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-'''19712/19683''', the '''symbiotic comma''', also known as the '''symbiosis''' or '''symbiosma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup]]) [[comma]] measuring about 2.55 [[cent]]s. It is the difference between an [[2187/2048|apotome (2187/2048)]] and an [[77/72|undecimal secor (77/72)]] – the latter of which can be broken down into a quartertone of [[33/32]] stacked on top of a third-tone of [[28/27]]. It is also the difference between a [[septischisma]] and a [[olympia]].
+'''19712/19683''', the '''symbiotic comma''', also known as the '''symbiosis''' or '''symbiosma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup]]) [[comma]] measuring about 2.55 [[cent]]s. It is the difference between a [[2187/2048|Pythagorean apotome (2187/2048)]] and an [[77/72|undecimal secor (77/72)]] – the latter of which can be broken down into a quartertone of [[33/32]] stacked on top of a third-tone of [[28/27]].
-It factors into the following commas:
+In terms of other commas, it is the difference between a [[septischisma]] and an [[olympia]]. It is also the sum of an [[argyria]] and an olympia, as well as the sum of a [[quartisma]] and a [[nexus comma]]. In the full 11-limit, it factors into simpler commas as ([[4375/4374]])⋅([[5632/5625]]), and in the 19-limit, ([[1216/1215]])⋅([[1540/1539]]).
-*[[2.3.7.11 subgroup]]:
-:* ([[olympia]])⋅([[argyria]])
-:* ([[quartisma]])⋅([[nexus comma]])
-* [[11-limit]]:
-:* ([[3025/3024]])⋅([[reef comma]])
-:*([[4375/4374]])⋅([[5632/5625]])
-* [[13-limit]]
-:* ([[676/675]])⋅([[chalmersia]])
-:* ([[1001/1000]])⋅([[phaotic comma]])
-:* ([[4096/4095]])⋅([[35035/34992]])
-* [[2.3.5.7.11.13.19 subgroup]]
-:* ([[1216/1215]])⋅([[1540/1539]])
 == Temperaments ==
-[[Tempering out]] this comma in the 11-limit results in the rank-4 '''symbiotic''' temperament (→ [[Rank-4 temperament #Symbiotic (19712/19683)]]), or in the 2.3.7.11 subgroup, the '''symbiotian''' temperament. In either case, it identifies [[11/8]] by an apotome over [[9/7]], [[11/6]] by an apotome over [[12/7]], and [[11/9]] by an apotome over [[8/7]]. Finally, it identifies 11/9 by a [[256/243|limma]] below 9/7.
+[[Tempering out]] this comma in the 11-limit results in the rank-4 '''symbiotic''' temperament (→ [[Rank-4 temperament #Symbiotic (19712/19683)]]), or in the 2.3.7.11 subgroup, the '''symbiotian''' temperament. In either case, 28/27 and 33/32 become apotome complements, and [[11/8]] is identified with an apotome over [[9/7]], [[11/6]] with an apotome over [[12/7]], and [[11/9]] with an apotome over [[8/7]]. Finally, 11/9 is identified with a [[256/243|limma]] below 9/7.
 === Symbiotian ===
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 [[Comma list]]: 19712/19683
-{{Mapping|legend=2| 1 0 0 -8 | 0 1 0 9 | 0 0 1 -1 }}
+{{Mapping|legend=3| 1 0 0 -8 | 0 1 0 9 | 0 0 1 -1 }}
 : mapping generators: ~2, ~3, ~7

Symbiotic comma: Difference between revisions

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